Automorphy for some \(l\)-adic lifts of automorphic mod \(l\) Galois representations. With Appendix A, summarizing unpublished work of Russ Mann, and Appendix B by Marie-France Vignéras.
type: Item
instance of: scholarly article
title: Automorphy for some \(l\)-adic lifts of automorphic mod \(l\) Galois representations. With Appendix A, summarizing unpublished work of Russ Mann, and Appendix B by Marie-France Vignéras.
zbMATH Open document ID: 1169.11020
author: Laurent Clozel
author: Michael Harris
author: Richard Taylor
published in: Publications Mathématiques
publication date: 2009-03-12T00:00:00Z
review text: The goal of the article under review is to extend the methods of \textit{A. Wiles} [Ann. Math. (2) 141, No. 3, 443--551 (1995; Zbl 0823.11029)] and \textit{R. Taylor, A. Wiles} [Ann. Math. (2) 141, No. 3, 553--572 (1995; Zbl 0823.11030)] from \(\mathrm{GL}_2\) to unitary groups of any rank. The authors work with the disconnected group \(\mathcal{G}_n\), which is the semidirect product of \(\mathrm{GL}_n\times \mathrm{GL}_1\) by the two element group. In this setting the Taylor-Wiles argument carries over well, and the authors prove \(R=T\) type of theorems in the ``minimal case'' (i.e., they consider deformation problems where the lifts on the inertia groups away from \(\ell\) are completely prescribed, e.g., as unramified as possible away from \(\ell\)). As the authors indicate in their very well-written introduction, there are three key inputs to their proof: (1) \textit{F. Diamond} [Invent. Math. 128, No. 2, 379--391 (1997; Zbl 0916.11037)] and K. Fujiwara's observation that Mazur's multiplicity one principle is not needed for the Taylor-Wiles argument. (2) A trick due to \textit{C. Skinner} and \textit{A. Wiles} [Duke Math. J. 107, No. 1, 15--25 (2001; Zbl 1016.11017)] which involves a base-change argument to bypass Ribet's level lowering results. (3) The proof of the local Langlands conjecture for \(\mathrm{GL}_n\) by the second and third named author, and its compatibility with the global Langlands correspondence. The authors prove the automorphy of certain Galois representations \(r: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \mathrm{GSp}_n(\mathbb{Z}_\ell)\) under a list of technical conditions. We do not state these hypotheses in detail, instead we give an overview of their content below. We refer the reader to Theorems 4.4.2 -- 4.4.3 and Corollary 4.4.4 of the paper under review. (\(H_I\)) The image of the representation \(\overline{r}:= r \bmod {\ell}\) restricted to \(\text{Gal}(\overline{F}/F(\zeta_\ell))\) should be big in the sense of De…
reviewed by: Kâzım Büyükboduk
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 11R34
zbMATH DE Number: 5528893
MaRDI profile type: Publication
full work available at URL: s10240-008-0016-1
OpenAlex ID: W4244408541
Wikidata QID: Q56341916
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DOI: 10.1007/S10240-008-0016-1
description: scientific article; zbMATH DE number 5528893